4 HUBERT SUM OF HUBERT SPACES 125

in E, and x = £ jn(xn) (observe that the series (jn(xn)) is not absolutely con-

n~l

vergent in general). This proves that the (algebraic) sum of the subspaces
E^ of E (which is obviously direct) is dense in E, in other words that the
smallest closed vector subspace containing all the E^ is E itself. Conversely:

(6.4.2) Let ¥ be a Hilbert space, (Fn) a sequence of closed subspaces such
that: (I) for m^n, any vector o/Fm is orthogonal to any vector of¥n; (2) the
algebraic sum H of the subspaces Fn is dense in F. Then, if E is the Hilbert
sum of the Fn, there is a unique isomorphism of F onto E which on each Fn
coincides with the natural injection jn of Fn into E.

Let F;, = jfB(FB), and let hn be the mapping of FB onto Fn, inverse to jn.
Let G be the algebraic sum of the Fn in E; that sum being direct, we can
define a linear mapping h of G into F by the condition that it coincides
with hn on each Fn. I claim that h is an isomorphism of G onto the prehilbert
space H (which, incidentally, will prove that the (algebraic) sum of the Fn
is direct in F); from the definition of the scalar product in E, we have to
check that

for xk e Ffc, yk e Fk; but by assumption (xh \ yk) = 0 if h ^ k, and the result
follows from the fact that eachyfc is an isomorphism. There is now a unique
continuous extension H of h which is a linear mapping of G = E into H = F,
by (5.5.4); the principle of extension of identities (3.15.2) and the con-
tinuity of the scalar product show that h is an isomorphism of E onto a
subspace of F, which, being complete and dense, must be F itself; the
inverse of E satisfies the conditions of (6.4.2). Its uniqueness follows from
the fact that it is completely determined in G and continuous in E (3.15.2).
Under the condition of (6.4.2), the Hilbert space F is often identified with
the Hilbert sum of its subspaces Fn.

Remark

(6.4.3) We can also prove (6.4.2) by establishing first that the sum of the Fn

n

is direct; indeed, if Z#j = 0 with jc£eFj (!</<TI) we also have

i=l

\xjI Z*«) - °for anyj^ n>and as (xjIxd = °for z ^•/»this boils down

\ i=l /is